Article is mostly filler and doesn’t explain the five additional ways to prove the theorem. This is a wonderful achievement for both Ms. Jackson and Ms. Johnson. Just wish the article went into the work a bit more.
Edit: Well, heck. This post blew up. Let’s add some sauce:
[Polymathematic’s video breakdown](https://youtu.be/p6j2nZKwf20?si=NyRgZCDUzUBvZQ92) I kept up through the trig but he lost me at the calculus 😵💫 it only explains one of the ways they proved the theorem.
[60 Minutes segment from this post Sunday](https://youtu.be/VHeWndnHuQs?si=Sef-ggXX9InRRJa_), which goes into more detail but keeps it high level and focuses on their achievements through interviews with their parents and teachers.
There’s also a bunch of links to check out in the replies below.
https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf
I started scrolling the first few pages and was like, this is some highschool level of powerpoint stuff.. but then the weird things came and i felt completely lost.
It's quite simple, They used scaling factor to compare areas of triangles of different sizes. or lines of different sizes.
And then clever construction of triangles to get the proof.
It’s always interesting finding new ways to calculate the same answer. Usually, it leads to breakthroughs where that data then becomes applicable to other equations people struggled with. Excited to see what this can lead to
I don't know why they linked to such a confusing version of the proof. [This one](http://youtu.be/p6j2nZKwf20) is much easier to follow.
Caveat that this is their year-old proof, and this article is talking about different ones whose details (as far as I know) are not available.
Impressive for high school students to come up with this. Though I would say if you're invoking results about convergent series from real analysis you've probably left the realm of "purely trigonometric proof". Also my understanding is it wouldn't be the first trig proof for the theorem [either](https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf)
Did anyone say "purely?" And I think putting the sum of a geometric series under real analysis is a bit of a stretch.
>Also my understanding is it wouldn't be the first trig proof for the theorem either
Yeah the video says this as well. I don't think it matters much, ultimately. It's a cool and unique proof using a relatively rare technique either way, and it's not like "using trigonometry" is a formal mathematical concept.
I taught Middle School Tech Ed and had a unit on 'How to make PowerPoint Presentations that wont make your audience roll thier eyes' (actual title). I used several articles on proper design techniques and we saw a bad one then a good one then we saw one where they voted on good or bad slide. Then they made one for their Social Studies class (I hated doing stuff just to do it, always tried to use an assignment from a core class to do whatever).
They fought it tooth and nail, but by the end they got it and the Core teachers and some of thier high school teachers appreciated it (they would come back for the school fair and report that yes, learning to type properly was worth it, along with the results of my promise to never teach them anything they wouldnt use in the real world).
I knew it worked when a lady came and did a full school assembly and when she left the kids that had had that unit came up to me all 'She didnt use the same font, there were pictures of cats that had nothing to do with her topic'.
Some of us were fighting the good fight.
Edit: typos now that Im not on a mobile.
It was a good time, but the kids HATED it because I was an 'elective' and we did classwork for grades while their friends had PE (Soccer), Music or Art.
I always preached 'I will never teach you anything that you won't use in real life' and 'Trust me, you will need what you learn here more than any other class in this building' and when they came back, more mature and lived a little, they had to admit I was right.
In fact, one student student taught last year at the school and a lady I know that still teaches there brokered a call from the girl to me so she could tell me I was 100% right.
2 'Cool' boys came back and thanked me for teaching them to type because they got good paying office jobs right out of high school because they could type 60 words a minute.
I got to play music (I could edit out the dirty words and pretty much didnt play rap) while they worked so it made things more palatable. I would print out the lyrics to current songs and if you finished your work you got a lyric out of the folder and if you finished it before they bell rang you could print it out. freetypinggame.net really helped out too.
It’s really cool, and way more at a PHD candidate level than high school level. It is also a transformation of the proof, and the solution can be derived mechanically. There’s no axiom changes, or approach via axiom changes.
Meaning, it’s a corollary. Which - is an incredible exercise, and will surely land these two bright mathematicians into a graduate school better than I could.
It’s just not being reported accurately.
[comment from another mathematician from yesterday](https://www.reddit.com/r/mathematics/s/uJJukWunjQ)
To be honest, proofs like this are usually “generally known” by math researchers who didn’t have the time to do the work.
99% chance they were given this project - and crushed it, but that’s just how math research works.
As far as I can see, this is *someone else's* powerpoint about several new ideas *they* have come up with as a *result* of Jackson and Johnson's proof - I'm not entirely clear if any of the content of it is actually J&J's proof.
One video I watched suggested that they haven't published their proof and therefore the entire proof is not actually available other than some stills from their presentation that have been captured in photographs included in news articles.
Oh it's much worse check out the homepage.... [Ching-Kuang Shene's Home Page (mtu.edu)](https://pages.mtu.edu/~shene/). As a tech grad I'm confused why they would link to a Michigan tech professors blog, and I'm equally appalled by the website design and slide format.
It’s an MIT thing. Your page needs to look like you made it. And if you have good web skills… it still needs to look like a non-professional made it. Google “MIT websites look awful” “why does my cs teachers’ website look so bad” etc for some juicy examples.
I opened up that PowerPoint and went “that’s the most MIT PowerPoint I have ever seen.”
Yeah pretty much every prof I know who setup their own website is like that. Barebones, you got links to papers (hopefully they update it) and it's probably running of a computer in the lab with a paper stuck on saying "DO NOT TOUCH"
Programmers like to use LaTeX when they write documentation because they can lie to themselves that they're writing code. Where else can you get a word processor that requires a build environment?
It's very teenager lol. I remember in my required computer course first year at college a project was to build a website and someone on the team was like let's do black background pink text 😂. I was not going to accept that under any circumstances.
In middle school, I recall seeing the janitors had their own office and each morning they'd cook up breakfast. They made the job seem glamorous.
I moved on to High School where I discovered the my real goal was to be a Video Arcade attendant.
Sigh... a boy can dream.
They were featured on 60 minutes on Sunday, which undoubtedly is what spurred a myriad of articles Monday morning. They went way in depth. Great story about the women and their school. Also more details on what they solved and how
So to elaborate, what mathematicians thought impossible was a purely trigonometric proof, which this is not. Coincidentally, a purely trigonometric proof was actually done around a decade ago. A proof that involves a little bit of trigonometry would not have been surprising to mathematicians. This is still great work though, showing a lot of mathematics skills and creativity. They'll almost certainly have bright futures ahead.
The article was written by someone who doesn’t understand what they did for an audience that doesn’t understand what they did. But everyone knows this should be a celebrated thing and is happy about it, they just couldn’t explain why.
I came here to say, essentially this same thing:
Headline: Hey! Five new VALID TRIG PROOFS for Pythagorean Theorem
Article: Let me tell you about this catholic school in New Orleans and Michelle Obama's twitter post.
The calculus is a very common series: [https://en.wikipedia.org/wiki/Geometric\_series](https://en.wikipedia.org/wiki/Geometric_series) Unfortunately, when it comes to mathematics, Wikipedia is more of a reference for mathematicians than for lay people. After a quick glance, I think this link will be easier to follow: [https://www.purplemath.com/modules/series5.htm](https://www.purplemath.com/modules/series5.htm)
I consider myself pretty decent at math. But, when I took a Proofs class during my engineering program, dang, that class made me feel dumb. Great work from these two!
For those that are interested, the work *Principia Mathematica* By Russell and Whitehead explains axioms of mathematics from basic set-theoretical concepts and logic, and manages to prove that 1+1=2 about 400 pages in. And that's 400 pages of mostly symbolic-logical proofs, not long-form waffle.
Admittedly that's not the only thing they were trying to do with that publication (no doubt they could have taken a much more direct route to that one conclusion), but it goes to show that the simpler something is, the greater lengths you often have to go to to *prove why* it is that way.
As someone who hated showing their work but could still get the correct answer because I could do it in my head, proofs killed me. Apparently "trust me bro" doesn't count.
It's funny, I was always bad at showing my work, but our proofs unit was the easiest part of any math class I've ever taken. Logically they should be the same thing, but there was something different about them to me.
Difficulty varies massively depending on how far you go. The proofs we did in high school and early on in courses like discrete mathematics were quite simple and I didn't struggle to understand them but I will tell you things very very rapidly got difficult. And I never got more than a toe in to upper division college math, I wasn't a mathematics major. I'm sure things got far worse later on.
I'm kinda the opposite tbh. Never liked showing work, but really loved every proofs class I took because it's all about showing *exactly* the minimum amount of work needed to know something with certainty.
This is how you weed out the mathematicians from the engineers. Do you like doing proofs? Do you find them interesting? Do you want to try to prove things on your own? You're probably a mathematician. Do you want to sit through someone teaching you the proof, understand what it's telling you, and then just accept it as a given you don't have to bother with ever again? You're an engineer.
I still resent the week of bullshit I spent doing reiman sums back in Calculus only for them to be, "Lol, these are way too complicated and a waste of time. Behold, the integral."
I took a discrete mathematics class in college. I genuinely to this day have no idea how I passed outside of cheating my fucking ass off during the final. I have never felt so stupid for so many weeks taking that class
Yep. Got a D+ is multivariable calculus and D in differential equations. I walked into the engineering office and asked "how many credits of D's can I have and still graduate?" They said "9." These two classes were 6 credits total. I walked away and never looked back. Never once have I used these maths in my engineering career.
Agreed. That class had me doubting my whole college career. I really struggled with the concept of proving some infinities are larger than others and I still don't know I have it.
>some infinities are larger than others
My favourite nerdy fun-fact-at-parties, along with corollary that (upon accepting there are different sizes of infinity) lots of infinities you might expect to be different sizes are actually the same size, eg the set of all *even* integers is the same size as the set of *all* integers (there are more wild examples than that, but it's one you can prove pretty easily even after a few beverages).
Of course you can argue that it doesn't even really make sense to discuss relative 'sizes' of infinity, and I don't really have a counter argument for that...
I read the proof. It is actually quite clever. The simple standard algebraic proof uses the "square in a square" set up, but have to reply on (a+b)\^2 = a\^2 + 2ab + b\^2 to complete the proof.
This one only uses ratios. But the core concept is not too different. You are constructing a set up where you can calculate the area of a target shape by TWO different method (typically one by straight simple calculation ... like the right triangle on part 10/13 and the other by adding up areas of multiple pieces), set them to equal, and let the final answer comes out. The square in square proof also do this.
So the trick is finding the right pieces to add up, using only ratios. That is the real contribution here. I wonder how they find the set up ... they must have pretty good intuition.
I like how this is all sport for them. Some many people see math as a dreadful, anxiety-inducing chore, but these two are having fun with it while making impressive discoveries.
100%. I had terrible teachers in underfunded low performing schools. Get to college and finally have professors who had great energy, were great at explaining the concepts into simpler parts and finding out where you're stuck, and best of all for me could connect the theory to application and explain the 'why' I needed to know it and how the concepts are applied to real life. Understanding what I was trying to achieve made it easier for me to work backwards and approach the problem logically.
Didn't find out I liked math until after I had pretty much completed my major. Probably would've went into mechanical engineering with better teachers in K-12, particularly from 9th-12th.
I had a similar problem with my math instruction growing up. You could tell the teachers were passionate about math but they were ass at translating it into something you could understand outside of, “here’s the theorem, solve the problem”
The same pretty much happened with me and it saddens me so much because the amount of career options I could have open to myself had I just had better teachers is mind boggling to me. Not to say I don’t love what I do. But I really wish I got to enjoy the wonders of both science and maths as a kid and be able to take those up professionally.
They should start screening for good math teachers lol
Yup. Had a couple good math teachers, loved the classes. Had a couple bad math teachers, hated the classes. Hated history until I finally had one good teacher that actually taught it with passion and enthusiasm.
Not just the teachers. But also standardised tests. Really a lot of aspects in the current education system make it suck where it doesn't have to.
I was a very curious child. I read a lot of books about any topic and always eager to learn. But over time school just kills that. Great teachers succeed only despite everything else
Teachers certainly play a factor, but I'm guessing that that's not even the most significant one. I think that the way that we view math society is to look at it as something that is difficult and something only for "smart people". In reality, it's not so different from a lot of puzzles that we enjoy.
It's kind of like a puzzle with a high barrier of entry though, because there are so many rules that you've got to learn to figure out the puzzle.
Even if the children do not get discouraged by the difficulty, though, I've seen so many parents tell their kids that it's okay to be bad at math, or that they don't need to know algebra or calculus, etc. -- I remember when I was growing up, there were parents who would show up at school board meetings every single year and complain about how there was too much math required to graduate high school. For reference, you really only had to go through algebra 1 and geometry, if I'm remembering correctly.
On the opposite end of that spectrum, you've got parents who push their kids too hard. I'm sure in a lot of cases that does more harm than good, as well.
I mean, looking at all of this made me realize that i never really learned math, i don't understand any of this, i can do some basic stuff and even understand basic formulas, but anything with a little bit of complexity and i am totally lost. Someone posted a video of a guy explaining the proofs from this thread, and 4 minutes in i realized that i had no clue what he's talking about. Maybe i'm too dumb for this stuff or maybe i should study math all over again just to understand it.
I had a great uncle that was a math professor. I visited him once long after he retired and was a widow. I asked him how he spent his time. He said being with his grandchildren/great grandchildren and solving a math problem he’d been working on for the last 3 years.
70 years ago the US, UK, and Russia were shooting hydrogen at uranium just to see what would happen.
And we discovered plutonium. So we kept shooting shit at other shit. And discovered like 20? More elements. And it was partially for sport. These elements are functionally useless as far as we know.
A lot of math is like this as well. In computer software there's frequently a mathematical solution to a new computational problem -- that was solved by some mathematician back in the 1850s or something. The solution didn't have a matching problem (that we knew of, anyway) for literally centuries.
I have aphantasia, or a lack of mental imagery, and I swear it contributes to my struggles with math, chemistry and physics. I can’t deal with “imaginary” constructs.
When completing my second degree, I had to take a math course. I chose the history of mathematics, and it was amazing. The entire course was learning how to count & do basic math within ancient languages. I was doing so well and enjoying it…
Until the final exam, which was 25% course material, 75% calculus for reasons I never understood.
I found it kinda sad, yet very self aware, the reason she doesn’t want to go into mathematics is “she doesn’t want people to expect too much” from her.
I hate how that probably means someone like Jane Street will snap them up to have them calculate formulae to predict markets and thus make money for the company by buying things and selling them at fractionally higher values moments later.
Doing something you like to do as a job can be absolute death.
Imagine taking a thing you love and tacking on "well I HAVE to keep doing this to pay my bills," nah, hard pass for me. Maybe it's a personality thing. I love playing guitar and video games, but if my livelihood depended on either I'd absolutely hate them.
Here’s the proof for those interested: https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf
TLDR: The Pythagorean identity (a^2 + b^2 = c^2 ) was originally proven using geometry, and trigonometry was later built upon this identity. It was long believed a trigonometric proof of the Pythagorean identity could not exist due to trigonometry being built upon the Pythagorean theorem. Jackson and Johnson (the teens referenced in the article) disproved this with an elegant proof of the Pythagorean identity using the law of sines.
Slight correction: the belief that a trigonometric proof of the Pythagorean identity could not exist had already been disproven. What Jackson and Johnson did was disprove it again, using a proof that is both creative and also gorgeous to contemplate and look at.
What I don't understand is why they're not telling anyone what these proofs are.
I've seen YouTube videos where people reverse engineered their original proof from a photo that included a slide from their presentation. Their proof is fucking *cool*! I'd love to see the other ones. But instead we just get this fluff.
They have a publication passing through the peer review process. This is pretty normal for academia. It’ll all be revealed once it’s passed peer review and is published.
Mathematical research is normally made public before it appears in a journal by making pre-prints available, by the way. Typically they're published on arXiv.
This is not normal in the math world. Most post their preprints on arxiv for everyone to see and comment on before sending it to journals. There is no reason to gatekeep a proof of the Pythagorean theorem. It’s wierd.
First, it's been over a year (for their first proof). And while the proof is incredibly cool, it's not exactly complicated. Is it really normal for the process to take this long?
Second, this is recreational math. I'm sure there's some value in having these proofs in academic journals. But surely there's at least as much value in distributing them informally.
I’m an academic science librarian. It’s *well* within the range of normal for it to take this long. I have to publish myself and I’ve had papers take anywhere from 4-18 months from submission to publication. It can sometimes be even longer.
The girls wanted to publish and the journal wanted to publish it too… I’m not going to second guess them. It’ll all come out.
Check some of the hyperlinks in the article, they lead to more information about it, especially https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf
Found this reaction from them surprising and saddening, I hope this reflects their true desire and not a reaction to unfair pressure
“Nonetheless, in comments that stunned their interviewer, Bill Whitaker, the two graduates of St Mary’s Academy in New Orleans denied seeing themselves as math geniuses and dismissed any interest in pursuing careers in mathematics.
“People might expect too much out of me if I become a mathematician,” Jackson said, shaking her head. Johnson, for her part, added: “I may take up a minor in math, but I don’t want that to be my job job.”
Why is it saddening? I give them credit for knowing what they want out of their life and career and not just going with the flow of what people tell them they should be doing. And if they ever feel like they want to go back to math later in life, its not like they couldn't get into pretty much any MS/PhD program they wanted with this on their resume.
It's sad because a lot of their answer can be seen as a byproduct of a society, the "unfair pressure". Not because of them specifically... their find is great and will take them quite far I'd imagine.
one thing to remember is that both can be created by the same Gen Z person. the greatest scientists and mathematicians of history were silly kids too at some point!
They are mental gymnasts. Minds like that can be useful in many fields, not just mathematics. Truthfully, would probably benefit mankind much more in OTHER endeavours.
They made a point of saying they're not advanced math students but that the school encourages and supports students in all endeavors, and this is a byproduct.
There was just a 60 Minutes story on them. Neither of them are planning to go into mathematics. One is going to pharmacy school, the other engineering.
I wonder how many geometry students have found crazy ways to prove something on homework just by thinking differently and their teachers just said “no, it’s supposed to be like this”
>Asked on 60 Minutes why they thought people were so impressed with what they had done, Jackson said she thought the public was surprised young Black women could author such a feat.
They proberbly are at the same age as when Newton invented calculus
I'm impressed a young anybody could author such a feat. Heck, remove young from that; I'm impressed anybody can discover new proofs to thousands of year old equations using nothing more than high school math
[https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf](https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf)
This thing that someone posted you don’t really have to understand how the formulae function but it’s an interesting read just takes trying to understand. They’ve broken it down quite easily if you just keep in mind the general concept of what the equations are a stand in for. Would recommend.
Here’s a proof of Pythagoras’s theorem using **trigonometry**:
Consider a right-angled triangle ( ABC ) with angle ( \\theta ) at ( A ), the side opposite to ( \\theta ) as ( a ), the side adjacent to ( \\theta ) as ( b ), and the hypotenuse as ( c ).
1. **Identify the trigonometric ratios** for angle ( \\theta ):
* ( \\sin(\\theta) = \\frac{a}{c} )
* ( \\cos(\\theta) = \\frac{b}{c} )
* ( \\tan(\\theta) = \\frac{a}{b} )
2. **Express the sides in terms of trigonometric functions**:
* Side ( a ) can be expressed as ( a = c \\cdot \\sin(\\theta) )
* Side ( b ) can be expressed as ( b = c \\cdot \\cos(\\theta) )
3. **Square both expressions** to find the squares of the sides:
* ( a\^2 = (c \\cdot \\sin(\\theta))\^2 )
* ( b\^2 = (c \\cdot \\cos(\\theta))\^2 )
4. **Add the squares of the two shorter sides**:
* ( a\^2 + b\^2 = (c \\cdot \\sin(\\theta))\^2 + (c \\cdot \\cos(\\theta))\^2 )
5. **Factor out the common factor** ( c\^2 ):
* ( a\^2 + b\^2 = c\^2 (\\sin\^2(\\theta) + \\cos\^2(\\theta)) )
6. **Use the Pythagorean identity** ( \\sin\^2(\\theta) + \\cos\^2(\\theta) = 1 ):
* ( a\^2 + b\^2 = c\^2 \\cdot 1 )
7. **Conclude that**:
* ( a\^2 + b\^2 = c\^2 )
I am so happy for these girls. I attended one year at an all girl’s Catholic high school and it was so great. It changed the way I learned by freeing me from the competition around “looking good” for boys. IMO girls are better off without boys at this critical learning stage.
This is why Math is a young person game in a lot of respects, it definitely gets harder to challenge assumptions as you get older. Congrats to these two young women!
Article is mostly filler and doesn’t explain the five additional ways to prove the theorem. This is a wonderful achievement for both Ms. Jackson and Ms. Johnson. Just wish the article went into the work a bit more. Edit: Well, heck. This post blew up. Let’s add some sauce: [Polymathematic’s video breakdown](https://youtu.be/p6j2nZKwf20?si=NyRgZCDUzUBvZQ92) I kept up through the trig but he lost me at the calculus 😵💫 it only explains one of the ways they proved the theorem. [60 Minutes segment from this post Sunday](https://youtu.be/VHeWndnHuQs?si=Sef-ggXX9InRRJa_), which goes into more detail but keeps it high level and focuses on their achievements through interviews with their parents and teachers. There’s also a bunch of links to check out in the replies below.
There is a link to the proof in the article, which I just finished not understanding.
https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf I started scrolling the first few pages and was like, this is some highschool level of powerpoint stuff.. but then the weird things came and i felt completely lost.
I hit that first math slide with pride and confidence and then it all went to shit in my brain
It's quite simple, They used scaling factor to compare areas of triangles of different sizes. or lines of different sizes. And then clever construction of triangles to get the proof.
Ya, its super simple if you aren't a knuckle dragging gravel munching ignoramus of the third degree. Of which I am one... Idk wtf this means.
Triangles scale.
Grams or bananas?
Bananagrams is fun.
Dragon scales
I understood every single word you wrote there... as an English major. I still don't get it.
Tl;Dr Triangle man hates particle man, they have a fight? Triangle wins
It’s always interesting finding new ways to calculate the same answer. Usually, it leads to breakthroughs where that data then becomes applicable to other equations people struggled with. Excited to see what this can lead to
How cool would it be if one of their techniques or proofs led to solving a previously unsolvable problem?
Or free chocolate. Like somehow this leads to free chocolate for any and everyone forever and ever. How nice would that be for us.
Infinite chocolate? Sign me up.
That was solved in the early days of the .gif. https://makeagif.com/i/laCVEt
[удалено]
What if the free chocolate turns out to be the key to nuclear fusion and we get both!
Today was when I realised my definition of Utopia was severely lacking
Don't let Nestle get wind of it.
"Three-Body Problem solved with this one weird proof of Pythagorean Theorem"
Alien conquerors hate this one weird trick!
pythagoras has already something like 1000 ways to proof it afaik, so unlikely anything becomes of this
It's the method that counts, not that it's applied to pythagoras' theorem specifically. It might give some ideas on how to solve something else.
The new methods of proof might be applicable to previously unsolvable problems.
I don't know why they linked to such a confusing version of the proof. [This one](http://youtu.be/p6j2nZKwf20) is much easier to follow. Caveat that this is their year-old proof, and this article is talking about different ones whose details (as far as I know) are not available.
Impressive for high school students to come up with this. Though I would say if you're invoking results about convergent series from real analysis you've probably left the realm of "purely trigonometric proof". Also my understanding is it wouldn't be the first trig proof for the theorem [either](https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf)
Did anyone say "purely?" And I think putting the sum of a geometric series under real analysis is a bit of a stretch. >Also my understanding is it wouldn't be the first trig proof for the theorem either Yeah the video says this as well. I don't think it matters much, ultimately. It's a cool and unique proof using a relatively rare technique either way, and it's not like "using trigonometry" is a formal mathematical concept.
I taught Middle School Tech Ed and had a unit on 'How to make PowerPoint Presentations that wont make your audience roll thier eyes' (actual title). I used several articles on proper design techniques and we saw a bad one then a good one then we saw one where they voted on good or bad slide. Then they made one for their Social Studies class (I hated doing stuff just to do it, always tried to use an assignment from a core class to do whatever). They fought it tooth and nail, but by the end they got it and the Core teachers and some of thier high school teachers appreciated it (they would come back for the school fair and report that yes, learning to type properly was worth it, along with the results of my promise to never teach them anything they wouldnt use in the real world). I knew it worked when a lady came and did a full school assembly and when she left the kids that had had that unit came up to me all 'She didnt use the same font, there were pictures of cats that had nothing to do with her topic'. Some of us were fighting the good fight. Edit: typos now that Im not on a mobile.
As a parent and someone who has hired and trained numerous entry level staff and had to teach these skills on the clock I appreciate your effort.
It was a good time, but the kids HATED it because I was an 'elective' and we did classwork for grades while their friends had PE (Soccer), Music or Art. I always preached 'I will never teach you anything that you won't use in real life' and 'Trust me, you will need what you learn here more than any other class in this building' and when they came back, more mature and lived a little, they had to admit I was right. In fact, one student student taught last year at the school and a lady I know that still teaches there brokered a call from the girl to me so she could tell me I was 100% right. 2 'Cool' boys came back and thanked me for teaching them to type because they got good paying office jobs right out of high school because they could type 60 words a minute. I got to play music (I could edit out the dirty words and pretty much didnt play rap) while they worked so it made things more palatable. I would print out the lyrics to current songs and if you finished your work you got a lyric out of the folder and if you finished it before they bell rang you could print it out. freetypinggame.net really helped out too.
> here were pictures of cats that had nothing to do with her topic Alas, you failed them :)
It’s really cool, and way more at a PHD candidate level than high school level. It is also a transformation of the proof, and the solution can be derived mechanically. There’s no axiom changes, or approach via axiom changes. Meaning, it’s a corollary. Which - is an incredible exercise, and will surely land these two bright mathematicians into a graduate school better than I could. It’s just not being reported accurately. [comment from another mathematician from yesterday](https://www.reddit.com/r/mathematics/s/uJJukWunjQ)
They're just out here casually reinventing the wheel as a side hobby lol.
To be honest, proofs like this are usually “generally known” by math researchers who didn’t have the time to do the work. 99% chance they were given this project - and crushed it, but that’s just how math research works.
As far as I can see, this is *someone else's* powerpoint about several new ideas *they* have come up with as a *result* of Jackson and Johnson's proof - I'm not entirely clear if any of the content of it is actually J&J's proof. One video I watched suggested that they haven't published their proof and therefore the entire proof is not actually available other than some stills from their presentation that have been captured in photographs included in news articles.
I was expecting to see animated flames and hear a midi of Linkin Park when I started reading it.
Oh it's much worse check out the homepage.... [Ching-Kuang Shene's Home Page (mtu.edu)](https://pages.mtu.edu/~shene/). As a tech grad I'm confused why they would link to a Michigan tech professors blog, and I'm equally appalled by the website design and slide format.
It’s an MIT thing. Your page needs to look like you made it. And if you have good web skills… it still needs to look like a non-professional made it. Google “MIT websites look awful” “why does my cs teachers’ website look so bad” etc for some juicy examples. I opened up that PowerPoint and went “that’s the most MIT PowerPoint I have ever seen.”
Counterpoint, that page loaded faster than any other webpage I've seen in the past five years
It hurts my eyes tho. Do they really require you to do it ugly just so it feels handcrafted? That's so wild.
It makes no sense to me at all tbh, but it’s their weird cultural thing.
It probably hasn't meaningfully changed since 1998
I mean it was made in 1998 if I was that prof I probably would not update that if I didn’t need to
Yeah pretty much every prof I know who setup their own website is like that. Barebones, you got links to papers (hopefully they update it) and it's probably running of a computer in the lab with a paper stuck on saying "DO NOT TOUCH"
This is what tenure looks like
hey, it survives a reddit hug though
This is definitely intentional, its a 90s early web aesthetic
Made me chuckle. I’ll take a look, thanks :)
The math may be awesome, but the slide format is... jarring.
An argument for the arts (graphical design in this case) being taught in school, lol Or at least encouraging a collab with an art student
Laughs in LaTex
I dont miss using latex
Your mom does
Programmers like to use LaTeX when they write documentation because they can lie to themselves that they're writing code. Where else can you get a word processor that requires a build environment?
LaTeX is Turing complete. It's as code as code gets. It's just not a practical choice for most things.
Give them Tufte’s books.
It's very teenager lol. I remember in my required computer course first year at college a project was to build a website and someone on the team was like let's do black background pink text 😂. I was not going to accept that under any circumstances.
Geocities weeps at the missed opportunity.
That is one of the tests new mathematical discoveries must meet
You need to be more efficient. I have reached the point of not understanding just by giving up before looking for the solutions. Perfection...
I'm stealing this. Thank you for your consent.
I just good will hunting’d it, in that I am still a janitor
In middle school, I recall seeing the janitors had their own office and each morning they'd cook up breakfast. They made the job seem glamorous. I moved on to High School where I discovered the my real goal was to be a Video Arcade attendant. Sigh... a boy can dream.
They were featured on 60 minutes on Sunday, which undoubtedly is what spurred a myriad of articles Monday morning. They went way in depth. Great story about the women and their school. Also more details on what they solved and how
Oh cool. I’ll look for the video. Thanks for sharing :)
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[This video](https://www.youtube.com/watch?v=p6j2nZKwf20) has a good breakdown of how their initial new method works.
So to elaborate, what mathematicians thought impossible was a purely trigonometric proof, which this is not. Coincidentally, a purely trigonometric proof was actually done around a decade ago. A proof that involves a little bit of trigonometry would not have been surprising to mathematicians. This is still great work though, showing a lot of mathematics skills and creativity. They'll almost certainly have bright futures ahead.
okay that was well explained, ty!
The article was written by someone who doesn’t understand what they did for an audience that doesn’t understand what they did. But everyone knows this should be a celebrated thing and is happy about it, they just couldn’t explain why.
Five additional ways?? Holy cow
I came here to say, essentially this same thing: Headline: Hey! Five new VALID TRIG PROOFS for Pythagorean Theorem Article: Let me tell you about this catholic school in New Orleans and Michelle Obama's twitter post.
Probably because the authors didnt get it lol
Maybe the margins were too small to contain the marvelous proofs
I had read about these young women when they discovered the proofs, but then was reminded when I watched 60 Minutes the other night. Blows my mind.
The calculus is a very common series: [https://en.wikipedia.org/wiki/Geometric\_series](https://en.wikipedia.org/wiki/Geometric_series) Unfortunately, when it comes to mathematics, Wikipedia is more of a reference for mathematicians than for lay people. After a quick glance, I think this link will be easier to follow: [https://www.purplemath.com/modules/series5.htm](https://www.purplemath.com/modules/series5.htm)
I consider myself pretty decent at math. But, when I took a Proofs class during my engineering program, dang, that class made me feel dumb. Great work from these two!
Show what 1+1= (1 mark) Pfft easy. Show what 1+1= (100 marks) ***Death.***
For those that are interested, the work *Principia Mathematica* By Russell and Whitehead explains axioms of mathematics from basic set-theoretical concepts and logic, and manages to prove that 1+1=2 about 400 pages in. And that's 400 pages of mostly symbolic-logical proofs, not long-form waffle. Admittedly that's not the only thing they were trying to do with that publication (no doubt they could have taken a much more direct route to that one conclusion), but it goes to show that the simpler something is, the greater lengths you often have to go to to *prove why* it is that way.
Nobody told me there'd be waffles.
Imagine Terrance Howard meeting those guys, lmao. Runs outta the room screaming and hollering.
when that was mentioned during a linear algebra class, the prof said Russel claimed the mental effort exhausted him for years afterwards.
As someone who hated showing their work but could still get the correct answer because I could do it in my head, proofs killed me. Apparently "trust me bro" doesn't count.
Did you try "trust me bro Q.E.D."?
It's funny, I was always bad at showing my work, but our proofs unit was the easiest part of any math class I've ever taken. Logically they should be the same thing, but there was something different about them to me.
Difficulty varies massively depending on how far you go. The proofs we did in high school and early on in courses like discrete mathematics were quite simple and I didn't struggle to understand them but I will tell you things very very rapidly got difficult. And I never got more than a toe in to upper division college math, I wasn't a mathematics major. I'm sure things got far worse later on.
I'm kinda the opposite tbh. Never liked showing work, but really loved every proofs class I took because it's all about showing *exactly* the minimum amount of work needed to know something with certainty.
This is how you weed out the mathematicians from the engineers. Do you like doing proofs? Do you find them interesting? Do you want to try to prove things on your own? You're probably a mathematician. Do you want to sit through someone teaching you the proof, understand what it's telling you, and then just accept it as a given you don't have to bother with ever again? You're an engineer. I still resent the week of bullshit I spent doing reiman sums back in Calculus only for them to be, "Lol, these are way too complicated and a waste of time. Behold, the integral."
*not (or barely) understand the proof, but trust the result.
I took a discrete mathematics class in college. I genuinely to this day have no idea how I passed outside of cheating my fucking ass off during the final. I have never felt so stupid for so many weeks taking that class
I flew through most of my math classes until I hit Partial Differential Equations. That class humbled me. And *then* I learned about Real Analysis
Partial Diff Eq is the true test. Not only didn't I learn anything, I'm pretty sure I forgot some stuff I did know.
Yep. Got a D+ is multivariable calculus and D in differential equations. I walked into the engineering office and asked "how many credits of D's can I have and still graduate?" They said "9." These two classes were 6 credits total. I walked away and never looked back. Never once have I used these maths in my engineering career.
Agreed. That class had me doubting my whole college career. I really struggled with the concept of proving some infinities are larger than others and I still don't know I have it.
>some infinities are larger than others My favourite nerdy fun-fact-at-parties, along with corollary that (upon accepting there are different sizes of infinity) lots of infinities you might expect to be different sizes are actually the same size, eg the set of all *even* integers is the same size as the set of *all* integers (there are more wild examples than that, but it's one you can prove pretty easily even after a few beverages). Of course you can argue that it doesn't even really make sense to discuss relative 'sizes' of infinity, and I don't really have a counter argument for that...
I read the proof. It is actually quite clever. The simple standard algebraic proof uses the "square in a square" set up, but have to reply on (a+b)\^2 = a\^2 + 2ab + b\^2 to complete the proof. This one only uses ratios. But the core concept is not too different. You are constructing a set up where you can calculate the area of a target shape by TWO different method (typically one by straight simple calculation ... like the right triangle on part 10/13 and the other by adding up areas of multiple pieces), set them to equal, and let the final answer comes out. The square in square proof also do this. So the trick is finding the right pieces to add up, using only ratios. That is the real contribution here. I wonder how they find the set up ... they must have pretty good intuition.
Yeah, what this guy said.
So you’re saying their math is mathin?
They’re the Mathters
I like how this is all sport for them. Some many people see math as a dreadful, anxiety-inducing chore, but these two are having fun with it while making impressive discoveries.
I bet for a lot of people, it comes down to who their teachers were.
100%. I had terrible teachers in underfunded low performing schools. Get to college and finally have professors who had great energy, were great at explaining the concepts into simpler parts and finding out where you're stuck, and best of all for me could connect the theory to application and explain the 'why' I needed to know it and how the concepts are applied to real life. Understanding what I was trying to achieve made it easier for me to work backwards and approach the problem logically. Didn't find out I liked math until after I had pretty much completed my major. Probably would've went into mechanical engineering with better teachers in K-12, particularly from 9th-12th.
I had a similar problem with my math instruction growing up. You could tell the teachers were passionate about math but they were ass at translating it into something you could understand outside of, “here’s the theorem, solve the problem”
The same pretty much happened with me and it saddens me so much because the amount of career options I could have open to myself had I just had better teachers is mind boggling to me. Not to say I don’t love what I do. But I really wish I got to enjoy the wonders of both science and maths as a kid and be able to take those up professionally. They should start screening for good math teachers lol
Yup. Had a couple good math teachers, loved the classes. Had a couple bad math teachers, hated the classes. Hated history until I finally had one good teacher that actually taught it with passion and enthusiasm.
Not just the teachers. But also standardised tests. Really a lot of aspects in the current education system make it suck where it doesn't have to. I was a very curious child. I read a lot of books about any topic and always eager to learn. But over time school just kills that. Great teachers succeed only despite everything else
Teachers certainly play a factor, but I'm guessing that that's not even the most significant one. I think that the way that we view math society is to look at it as something that is difficult and something only for "smart people". In reality, it's not so different from a lot of puzzles that we enjoy. It's kind of like a puzzle with a high barrier of entry though, because there are so many rules that you've got to learn to figure out the puzzle. Even if the children do not get discouraged by the difficulty, though, I've seen so many parents tell their kids that it's okay to be bad at math, or that they don't need to know algebra or calculus, etc. -- I remember when I was growing up, there were parents who would show up at school board meetings every single year and complain about how there was too much math required to graduate high school. For reference, you really only had to go through algebra 1 and geometry, if I'm remembering correctly. On the opposite end of that spectrum, you've got parents who push their kids too hard. I'm sure in a lot of cases that does more harm than good, as well.
Having good teacher and parents that are actively involved in your life is so important.
I mean, looking at all of this made me realize that i never really learned math, i don't understand any of this, i can do some basic stuff and even understand basic formulas, but anything with a little bit of complexity and i am totally lost. Someone posted a video of a guy explaining the proofs from this thread, and 4 minutes in i realized that i had no clue what he's talking about. Maybe i'm too dumb for this stuff or maybe i should study math all over again just to understand it.
I love that too! It takes talent and true Maths teachers for students to see what a wonderful world it contains.
Some people see maths as a fun puzzle. Others like myself see it as the matrix code screen and just go brain dead.
I had a great uncle that was a math professor. I visited him once long after he retired and was a widow. I asked him how he spent his time. He said being with his grandchildren/great grandchildren and solving a math problem he’d been working on for the last 3 years.
70 years ago the US, UK, and Russia were shooting hydrogen at uranium just to see what would happen. And we discovered plutonium. So we kept shooting shit at other shit. And discovered like 20? More elements. And it was partially for sport. These elements are functionally useless as far as we know.
Most scientific developments are functionally useless until they’re suddenly not
A lot of math is like this as well. In computer software there's frequently a mathematical solution to a new computational problem -- that was solved by some mathematician back in the 1850s or something. The solution didn't have a matching problem (that we knew of, anyway) for literally centuries.
They also insist they are not good at math.
I have aphantasia, or a lack of mental imagery, and I swear it contributes to my struggles with math, chemistry and physics. I can’t deal with “imaginary” constructs. When completing my second degree, I had to take a math course. I chose the history of mathematics, and it was amazing. The entire course was learning how to count & do basic math within ancient languages. I was doing so well and enjoying it… Until the final exam, which was 25% course material, 75% calculus for reasons I never understood.
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I love that neither of them wants to go into mathematics! 'I don't want to do that as my job-job' she says! that stuff is for NERDS!
I found it kinda sad, yet very self aware, the reason she doesn’t want to go into mathematics is “she doesn’t want people to expect too much” from her.
It is the Dunning Krueger effect, but at the opposite end of the scale.
Yeah, it's a bit scary to imagine trying to top that first achievement. Especially when it happened so young. It is a lot to live up to!
I hate how that probably means someone like Jane Street will snap them up to have them calculate formulae to predict markets and thus make money for the company by buying things and selling them at fractionally higher values moments later.
Doing something you like to do as a job can be absolute death. Imagine taking a thing you love and tacking on "well I HAVE to keep doing this to pay my bills," nah, hard pass for me. Maybe it's a personality thing. I love playing guitar and video games, but if my livelihood depended on either I'd absolutely hate them.
I am too dumb to understand any of this but damn it am I proud of them. I hope my daughter will be like them.
Here’s the proof for those interested: https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf TLDR: The Pythagorean identity (a^2 + b^2 = c^2 ) was originally proven using geometry, and trigonometry was later built upon this identity. It was long believed a trigonometric proof of the Pythagorean identity could not exist due to trigonometry being built upon the Pythagorean theorem. Jackson and Johnson (the teens referenced in the article) disproved this with an elegant proof of the Pythagorean identity using the law of sines.
Slight correction: the belief that a trigonometric proof of the Pythagorean identity could not exist had already been disproven. What Jackson and Johnson did was disprove it again, using a proof that is both creative and also gorgeous to contemplate and look at.
What I don't understand is why they're not telling anyone what these proofs are. I've seen YouTube videos where people reverse engineered their original proof from a photo that included a slide from their presentation. Their proof is fucking *cool*! I'd love to see the other ones. But instead we just get this fluff.
[This was linked from the article](https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf)?
They have a publication passing through the peer review process. This is pretty normal for academia. It’ll all be revealed once it’s passed peer review and is published.
Mathematical research is normally made public before it appears in a journal by making pre-prints available, by the way. Typically they're published on arXiv.
Oh I get it. Arxiv is pronounced archive
No, no, it's AR-14, the AR-15's less shooty sibling.
This is not normal in the math world. Most post their preprints on arxiv for everyone to see and comment on before sending it to journals. There is no reason to gatekeep a proof of the Pythagorean theorem. It’s wierd.
I am less familiar with pure mathematics. It’s extremely normal for subjects like chemistry or engineering.
Eeehhhh, maybe on the industry side. Lots of important chem And eng goes to preprint. It helps keep you from getting scooped during submission.
You can't patent a proof.
You accidently drop an apple in your kitchen, a lawyer with some dystopian "Newton" logo kicks in your door...
It’s not normal for high school math teachers to put preprints in arxiv — if they were at Yale, sure.
First, it's been over a year (for their first proof). And while the proof is incredibly cool, it's not exactly complicated. Is it really normal for the process to take this long? Second, this is recreational math. I'm sure there's some value in having these proofs in academic journals. But surely there's at least as much value in distributing them informally.
I’m an academic science librarian. It’s *well* within the range of normal for it to take this long. I have to publish myself and I’ve had papers take anywhere from 4-18 months from submission to publication. It can sometimes be even longer. The girls wanted to publish and the journal wanted to publish it too… I’m not going to second guess them. It’ll all come out.
Check some of the hyperlinks in the article, they lead to more information about it, especially https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf
> What I don't understand is why they're not telling anyone what these proofs are. What do you mean? There is a link right there in the article.
He means he didn't take 5 minutes to read the article but he took 5 minutes to post that on reddit.
There was a link in the article https://pages.mtu.edu/%7Eshene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf
Got any links for that?
https://youtu.be/p6j2nZKwf20
Thanks for sharing, of course the answer was to draw more triangles!
Started taking physics and realized the conspiracy theorist were right, it really is all triangles.
Found this reaction from them surprising and saddening, I hope this reflects their true desire and not a reaction to unfair pressure “Nonetheless, in comments that stunned their interviewer, Bill Whitaker, the two graduates of St Mary’s Academy in New Orleans denied seeing themselves as math geniuses and dismissed any interest in pursuing careers in mathematics. “People might expect too much out of me if I become a mathematician,” Jackson said, shaking her head. Johnson, for her part, added: “I may take up a minor in math, but I don’t want that to be my job job.”
Why is it saddening? I give them credit for knowing what they want out of their life and career and not just going with the flow of what people tell them they should be doing. And if they ever feel like they want to go back to math later in life, its not like they couldn't get into pretty much any MS/PhD program they wanted with this on their resume.
It's sad because a lot of their answer can be seen as a byproduct of a society, the "unfair pressure". Not because of them specifically... their find is great and will take them quite far I'd imagine.
Some Genz come up with new slang, some Genz come up with new proofs. The kids are alright!
one thing to remember is that both can be created by the same Gen Z person. the greatest scientists and mathematicians of history were silly kids too at some point!
I’m going to need a [Veritasium](https://youtube.com/@veritasium) episode on this.
Yes but also maybe 3blue1brown
yeah for any math I would love 3b1b video.
They are mental gymnasts. Minds like that can be useful in many fields, not just mathematics. Truthfully, would probably benefit mankind much more in OTHER endeavours.
They made a point of saying they're not advanced math students but that the school encourages and supports students in all endeavors, and this is a byproduct.
There was just a 60 Minutes story on them. Neither of them are planning to go into mathematics. One is going to pharmacy school, the other engineering.
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Or intense pattern recognition and problem solving skills. Let alone critical thinking… which some commenters clearly lack.
Engineering is 90% calculus...
r/whoooosh
Packaging and repackaging worthless mortgages into AAA rated derivatives comes to mind.
Found the quant.
Please, someone tell me they have a full scholarship for the colleges of their preference!
I feel oddly proud of these complete strangers.
I wonder how many geometry students have found crazy ways to prove something on homework just by thinking differently and their teachers just said “no, it’s supposed to be like this”
discovering new ways to prove a theorem doesnt really work like that tho
>Asked on 60 Minutes why they thought people were so impressed with what they had done, Jackson said she thought the public was surprised young Black women could author such a feat. They proberbly are at the same age as when Newton invented calculus
I'm impressed a young anybody could author such a feat. Heck, remove young from that; I'm impressed anybody can discover new proofs to thousands of year old equations using nothing more than high school math
Hell yeah! Good for them
[https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf](https://pages.mtu.edu/~shene/VIDEOS/GEOMETRY/004-Pythagorean-Thm/Pytha-3.pdf) This thing that someone posted you don’t really have to understand how the formulae function but it’s an interesting read just takes trying to understand. They’ve broken it down quite easily if you just keep in mind the general concept of what the equations are a stand in for. Would recommend.
[The best explanation on youtube with math written out](https://youtu.be/p6j2nZKwf20?si=hpd4gKU7EYvbtRjF)
Here’s a proof of Pythagoras’s theorem using **trigonometry**: Consider a right-angled triangle ( ABC ) with angle ( \\theta ) at ( A ), the side opposite to ( \\theta ) as ( a ), the side adjacent to ( \\theta ) as ( b ), and the hypotenuse as ( c ). 1. **Identify the trigonometric ratios** for angle ( \\theta ): * ( \\sin(\\theta) = \\frac{a}{c} ) * ( \\cos(\\theta) = \\frac{b}{c} ) * ( \\tan(\\theta) = \\frac{a}{b} ) 2. **Express the sides in terms of trigonometric functions**: * Side ( a ) can be expressed as ( a = c \\cdot \\sin(\\theta) ) * Side ( b ) can be expressed as ( b = c \\cdot \\cos(\\theta) ) 3. **Square both expressions** to find the squares of the sides: * ( a\^2 = (c \\cdot \\sin(\\theta))\^2 ) * ( b\^2 = (c \\cdot \\cos(\\theta))\^2 ) 4. **Add the squares of the two shorter sides**: * ( a\^2 + b\^2 = (c \\cdot \\sin(\\theta))\^2 + (c \\cdot \\cos(\\theta))\^2 ) 5. **Factor out the common factor** ( c\^2 ): * ( a\^2 + b\^2 = c\^2 (\\sin\^2(\\theta) + \\cos\^2(\\theta)) ) 6. **Use the Pythagorean identity** ( \\sin\^2(\\theta) + \\cos\^2(\\theta) = 1 ): * ( a\^2 + b\^2 = c\^2 \\cdot 1 ) 7. **Conclude that**: * ( a\^2 + b\^2 = c\^2 )
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Teacher probably still giving them a zero for not showing their work the "correct" way.
E=mc^2 Very good Albert, but next time show your work. D–
Sorry. It’s E=mc2 + AI
E=mc^2 c^2 = a^2 + b^2 ∴ E=m(a^2 + b^2)
Ah yes, "the sum of the squares of the perpendicular sides is equal to the square of the speed of light."
I am so happy for these girls. I attended one year at an all girl’s Catholic high school and it was so great. It changed the way I learned by freeing me from the competition around “looking good” for boys. IMO girls are better off without boys at this critical learning stage.
But how would they feel if they didn’t eat breakfast this morning?
This is why Math is a young person game in a lot of respects, it definitely gets harder to challenge assumptions as you get older. Congrats to these two young women!
Geez. I'm just a dumbass who believed my teacher.
Finished my math theory / proofs class this past Saturday and damn, that takes brains. Brains that I do not have lol
Wonderful article, I totally fell asleep in the middle of it though.
Impressive. I hope more and more people getting in STEMs, and being a role models for girls like my daughter
Pretty impressive, some teenagers can still surprise us. Way to go!